Dr Ian Plummer
Technical
Ball Bounce Testing
What is required is a convenient and cheap method of testing the bounce (related
to the coefficient of restitution) of croquet balls. It should be accurate,
reproducible and preferably absolute. If not absolute then there should be
an inexpensive means of calibrating the measurement.
The coefficient of restitution C, is the ratio of the difference in velocities
before and after collision.
C = v_{f }/v_{i}, where v_{i} is the initial velocity
(difference) and v_{f} is the final velocity (difference).
The impact force at which the ball is tested should lie within those encountered
in average strokes played in croquet. The test should allow the bounce from
a specific region of a ball to be recorded so that variation over the surface
can be determined. Finally there should be no damage to the ball.
Basics
In the methods conceived so far energy must be provided to the ball or a proofing
surface which then collide. The parameters at collision or the results thereafter
are recorded and analysed to yield a 'bounce height'. The source of energy
conventionally has been potential energy derived from allowing a weight to
fall a measured distance under gravity. This has the advantage that it is cheap,
simple and reproducible. The height is a measure of Kinetic Energy, which is
related to the square of the velocity.
E_{k} = 1/2( m.v^{2 })
Methods to date give the coefficient of restitution in terms of a bounce height
under standard conditions: for a ball dropped from 60" it should bounce to
a height between 3045". When dropped from h_{i} = 60" it has a total
initial energy E_{i} of:
m.g.h_{i} = E_{i} = 1/2( m.v_{i}^{2})
m = mass of ball, g is accelleration due to gravity (9.8 m/s) and h the height.
After the collision (bounce) it reaches a new height h_{f}:
m.g.h_{f} = E_{f} = 1/2( m.v_{f}^{2}),
hence
C = sqrt( h_{f }/ h_{i} )
Hence a rebound range of 5075% yields a coefficient restitution of sqrt(0.50)
to sqrt(0.75) [= 0.707 ... 0.866]. The reduction in bounce height is a consequence
of energy losses on collision, due to deformation and heating of the ball and
energy transfer to the metal plate.
Methods
Steel Plate Drop Rig
The steel plate rig is described elsewhere. It consists of a 1" steel plate
set solidly in concrete onto which a ball is dropped and the rebound height
is recorded. This has been found to be sensitive to the details of the steel
to concrete bond, the size of the concrete block and the seating of the block
on the ground. Variations of 5% have been documented.
Calibrated Steel Plate Rig
As indicated above the problem with the steel plate rig is that each rig will
have its own variaion due to its construction. One method of obviating
this is to calaibrate the rig. It is not viable to send calibrated balls
around the world hence a convenient calibration standard is required. Suggestions
to date have been to use a large steel ball bearing or a child's large marble
(solid glass sphere). This would be dropped from a standard height and
its rebound height used to calibrate the rig. It then produces a coefficent
by which to modify the measured results on that rig.
Pendulum Impact
In
order to remove the problems of the concrete block from the measurement, a
pendulum method is proposed here. A ball attached to the end of a piece of
string is allowed to fall in an arc and hit a metal plate similarly suspended
from the same pivot point (see diagram). The initial energy given to the ball
is a function of the height from which it is dropped. The moving ball strikes
the stationary plate and both rebound. The energy of each is a function of
the height that they rebound to. Three measurements are needed, together with
the mass of the ball and plate: the initial and final height of the ball and
the rebound height of the plate.
The mass multiplied by the height times a constant (g  the gravitational
constant) gives the energy of that object. In a perfect system, with no losses,
the initial energy of the ball due to its height would be partitioned into
the steel plate and ball.
m_{b}*hi_{b} * g = (m_{b}*hf_{b} + m_{p}*h_{p})*g
+ L
where
m_{b} 
mass of ball 
hi_{b} 
initial height from which ball is dropped relative
to bottom of arc 
g 
graviational constant (9.81 ms^{1}) 
hf_{b} 
final maximum height of ball after collision
relative to bottom of arc 
m_{p} 
mass of plate 
h_{p} 
final maximum height of plate after collision
relative to bottom of arc 
L 
energy losses 
As an initial approximation the energy loss is due to the coefficient of restitution,
C. This yields:
C = sqrt [(m_{b}*hf_{b} + m_{p}*h_{p})/m_{b}*hi_{b}]
More accurately the righthand side is equal to the energy losses divided
by g. Addition energy losses, which should not be incorporated into the coefficient
of restitution, include the air resistance of the two bodies and any normal
modes (vibrations) excited in the plate. The design of the rig should be such
that the plate and ball gain no angular momentum in the collision. In practice
all that is required is the masses of the ball and plate and a measurement
of the drop height and the heights reached by the ball on the plate on rebound.
Practical details
To
reproduce the same impact collisions as used in the conventional steel plate
test the ball velocity can be matched by allowing it to fall (in an arc) through
the same vertical distance as for the plate test. Thus the ball centrepivot
distance should be 60". The mass of the plate is a user variable. If too small
it is difficult to measure the rebound (or indeed follow through) of the ball,
and if too large then the deflection of the plate is too small to measure accurately.
A mass in the order of 0.5  1.5 times the mass of the ball (454gm, 16oz) would
be appropriate.
The measurement to the centre of the ball (equivalent to its centre of mass)
from the pivot should be 60". Similarly the same measurement is used for the
plate. The ball must strike the plate at its centre of mass otherwise torques
(rotation) will be introduced. These will make the deflection difficult to
measure and abstract energy. It is therefore proposed that the plate is suspended
from two strings lying in the same plane as the pivot and centre of mass of
the plate at right angles to the motion of the ball. (See diagram). Similarly
the ball can be in a cradle supported by two similar strings. The masses of
the strings should be negligible in comparison to the plate or ball masses.
The cradle will allow different part s of the surface of the test ball to strike
the plate allowing the variation of bounce with position to be measured.
When measuring deflections it is the height of the centres of masses which
needs to be recorded.
Thanks to Nick Furze for spotting a missing sqrt.
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